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<p><dfn class="terminology">Solution:</dfn></p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_29.html">
\begin{equation*}
\begin{aligned}
&amp;M(x, y)=y \cos x+2 x e^y,\quad N(x, y)=\sin x+x^2 e^y-1,\\
&amp;\frac{\partial M}{\partial y}=\cos x+2 x e^y,\quad \frac{\partial N}{\partial x}=\cos x+2 x e^y,~\rightarrow \frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}.
\end{aligned}
\end{equation*}
</div>
<p class="continuation">Therefore, the ODE is exact. Thus there exists a <span class="process-math">\(\Psi(x, y)\)</span> such that</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_29.html">
\begin{equation}
\frac{\partial \Psi(x, y)}{\partial x}=M(x, y)=y \cos x+2 x e^y,\quad \frac{\partial \Psi(x, y)}{\partial y}=N(x, y)=\sin x+x^2 e^y-1.\tag{2.6.5}
\end{equation}
</div>
<p class="continuation">Integrating <span class="process-math">\((\ref{eq2_28})_1\text{,}\)</span> we obtain</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_29.html">
\begin{equation*}
\Psi(x, y)=y\sin x+x^2 e^y+h(y).
\end{equation*}
</div>
<p class="continuation">Further, from <span class="process-math">\((\ref{eq2_28})_2\text{,}\)</span> one has</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_29.html">
\begin{equation*}
\begin{aligned}
&amp;\frac{\Psi(x, y)}{\partial y}=\sin x+x^2 e^y+\frac{\textrm{d} h(y)}{\textrm{d} y}=N(x, y)=\sin x+x^2 e^y-1\\
&amp;~\rightarrow~\frac{\textrm{d} h(y)}{\textrm{d} y}=-1~\rightarrow~h(y)=-y+C_1~.
\end{aligned}
\end{equation*}
</div>
<p class="continuation">Setting <span class="process-math">\(C_1=0\text{,}\)</span> we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_29.html">
\begin{equation*}
\Psi(x, y)=y\sin x+x^2 e^y+h(y)=y\sin x+x^2 e^y-y.
\end{equation*}
</div>
<p class="continuation">Hence the solution to (<a href="" class="xref" data-knowl="./knowl/eq2_29.html" title="Equation 2.6.4">(2.6.4)</a>) is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_29.html">
\begin{equation*}
y\sin x+x^2 e^y-y=C.
\end{equation*}
</div>
<span class="incontext"><a href="sec2_6.html#p-48" class="internal">in-context</a></span>
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